On the omnipresence of sincere partial tilting and/or cotilting modules

We describe the aboundance of selforthgonal modules big enough to satisfy all "local" properties of tilting and/or cotilting modules, together with a "global" and functorial Hom-Ext condition, concerning the Kernels of certain functors, namely their intersection. One of the reasons of the big gap between classical and non classical partial tilting and/or cotilting modules seems to be the "unessential" role often played by projective (resp. injective) summands of tilting (resp. cotilting) modules M of projective (resp. injectve) dimension at most two. Even in the larger world of left bounded complexes (with finitely generated projective components) there is a big gap between titing complexes, in the sense of Rickard, and complexes, obtained from partial tilting modules with a nice and functorial behaviour. Indeed these last complexes may have a very easy structure,and, by repeating the remark of M. Shaps and E. Zakaj-Illouz (in "Combinatorial Partial Tilting Complexes for the Brauer Star Algebras", Marcel Dekker, volume 24, 2002, 187-207) on the structure of their indecomposable "elementary complexes", these complexes may have a very "combinatorial character".